The Hubbard model [1]} has come to play a paradigmatic role in the theory of electronic correlations in quantum materials where interactions play an essential role. Its centrality in the quantum statistical mechanics of interacting fermions is comparable to that of the Ising model in classical statistical mechanics, yet almost sixty years after it was first described, there remain unsettled basic questions, even concerning the actual phases that arise for various lattice geometries.
Conversely, the more progress that is made in obtaining theoretical solutions, the clearer it becomes that this simple model can exhibit a startling array of phases and regimes, many of which have clear parallels with observed behaviors of a wide variety of complex materials. For instance, there is compelling evidence that ferromagnetism, various forms of antiferromagnetism, unconventional superconductivity, charge-density waves, electronic liquid crystalline phases, and topologically ordered phases (e.g., “spin liquids”), among other phases, occur in specific realizations of the Hubbard model.
